1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

12.6. ELIMINATING As ON THE PERMUTATION MODULE S27 remains to establish (2), we may assume that L is a block. Let a := g-^1 Noti ...

S2S 12. LARGER GROUPS OVER Fz IN .Cj(G, T) We are now in a position to complete the proof of Theorem 12.6.2. By 12.6.18.2, Kv is ...

12.6. ELIMINATING As ON THE PERMUTATION MODULE S29 (2) Vg E vca(v) for each v E V n V^9. (3) r(G, V) :'.'.: 3, and r(G, V) :'.'. ...

830 12. LARGER GROUPS OVER F2 IN .Cj (G, T) PROOF. Assume Zv = Cv(L) -=/= 1. Let HE 1-l*(T, M), K := 02 (H), VH := (zH), and H* ...

12.6. ELIMINATING As ON THE PERMUTATION MODULE 831 We next prove (7). Most parts of Hypothesis F.9.1 are easy to check: For exam ...

832 i2. LARGER GROUPS OVER F2 IN .Cj(G,T) and Vtf, so from the structure of 6 := Out(U1) ~ ot(2), i::::: Ca(Li) ~ 83 x 83. But n ...

12.6. ELIMINATING A 8 ON THE PERMUTATION MODULE 833 Therefore m3(K) = 1. As m(P) = 2 = m 3 (PK), we conclude that PK:= PnK is of ...

834 12. LARGER GROUPS OVER F2 IN .Cj(G, T) PROOF. Recall Mv acts as A 8 or 8s on the set 0 of eight points. Thus if I is an invo ...

i2.6. ELIMINATING As ON THE PERMUTATION MODULE 835 that Cv(Ao) = VnNa(VY) for all such A 0. That is, A E A 3 -k(Mv, V). Therefor ...

836 12. LARGER GROUPS OVER F2 IN .Cj(G, T) PROOF. By 12.6.30.3 we may apply F.9.18 to conclude that K /Z(K) is a Bender group, L ...

12.6. ELIMINATING As ON THE PERMUTATION MODULE 837 This is a contradiction as T permutes with no nontrivial p-subgroups of L for ...

838 i2. LARGER GROUPS OVER F2 IN Cj (G, T) Suppose now that [Ui, K2] = 1. Then Vs = V5,i EEl Vs,2, where V5,i := [il5, Xi] ~ E 4 ...

12.7. THE TREATMENT OF Aa ON A 6-DIMENSIONAL MODULE 839 12.2.13 until after both He and M 2 4 have been independently identified ...

840 12. LARGER GROUPS OVER Fz IN .Cj(G, T) F-hyperplane of F V. Hence as Ri, i = 1, 2, are representatives for the conjugacy cla ...

12.7. THE TREATMENT OF As ON A 6-DIMENSIONAL MODULE 841 x EX, Kt"' = K'f:::; Ca(vt"') = Ca(Vt):::; Gt, so that Kt= Kt"'· Hence X ...

842 12. LARGER GROUPS OVER F2 IN L'.j(G, T) Qz = P :::;! Gz, and thenasT/P 95: Ds is Sylow in Gz/P::; ot(2) 95: Ss, we conclude ...

i2.7. THE TREATMENT OF A 6 ON A 6-DIMENSIONAL MODULE 843 by (4). Finally if 2 E 02, we may take 2 to act on Vt by (3), so 1 =/:- ...

844 12. LARGER GROUPS OVER F2 IN .C'f (G, T) LEMMA 12.7.16. If U = R1, then G ~ M24. PROOF. Assume that U = R 1. By 12.7.2.2, [U ...

12.7. THE TREATMENT OF A 6 ON A 6-DIMENSIONAL MODULE S45 Let a+ := M 24. Arguing as in the proof of Theorem 12.7.7, Mis determin ...

S46 i2. LARGER GROUPS OVER F2 IN Cj(G, T) so (P n UB)+ = Gp+(x). Recall X = 02 (0 2 ,z(L)), so X::;; L::;; I::;; Na(P). LEMMA 12 ...